In this article we will look at linear function, graph of a linear function and its properties. And, as usual, we will solve several problems on this topic.

Linear function called a function of the form

In a function equation, the number we multiply by is called the slope coefficient.

For example, in the function equation ;

in the equation of the function ;

in the equation of the function ;

in the function equation.

The graph of a linear function is a straight line.

1. To plot a function, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the function equation, and use them to calculate the corresponding y values.

For example, to plot a function graph, it is convenient to take and , then the ordinates of these points will be equal to and .

We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function:

2 . In a function equation, the coefficient is responsible for the slope of the function graph:

Title="k>0">!}

The coefficient is responsible for shifting the graph along the axis:

Title="b>0">!}

The figure below shows graphs of functions; ;

Note that in all these functions the coefficient greater than zero right. Moreover, the higher the value, the steeper the straight line goes.

In all functions - and we see that all graphs intersect the OY axis at point (0;3)

Now let's look at the graphs of functions; ;

This time in all functions the coefficient less than zero, and all function graphs are sloped left.

Note that the larger |k|, the steeper the straight line. The coefficient b is the same, b=3, and the graphs, as in the previous case, intersect the OY axis at point (0;3)

Let's look at the graphs of functions; ;

Now the coefficients in all function equations are equal. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:

The graph of the function (b=3) intersects the OY axis at point (0;3)

The graph of the function (b=0) intersects the OY axis at the point (0;0) - the origin.

The graph of the function (b=-2) intersects the OY axis at point (0;-2)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function looks like.

If k<0 и b>0 , then the graph of the function looks like:

If k>0 and b>0 , then the graph of the function looks like:

If k>0 and b<0 , then the graph of the function looks like:

If k<0 и b<0 , then the graph of the function looks like:

If k=0 , then the function turns into a function and its graph looks like:

The ordinates of all points on the graph of the function are equal

If b=0, then the graph of the function passes through the origin:

This direct proportionality graph.

3. I would like to separately note the graph of the equation. The graph of this equation is a straight line parallel to the axis, all points of which have an abscissa.

For example, the graph of the equation looks like this:

Attention! The equation is not a function, since different values ​​of the argument correspond to the same value of the function, which does not correspond.

4 . Condition for parallelism of two lines:

Graph of a function parallel to the graph of the function, If

5. The condition for the perpendicularity of two straight lines:

Graph of a function perpendicular to the graph of the function, if or

6. Points of intersection of the graph of a function with the coordinate axes.

With OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero in the equation of the function instead of x. We get y=b. That is, the point of intersection with the OY axis has coordinates (0; b).

With OX axis: The ordinate of any point belonging to the OX axis is equal to zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero in the equation of the function instead of y. We get 0=kx+b. From here. That is, the point of intersection with the OX axis has coordinates (;0):

Let's look at problem solving.

1. Construct a graph of the function if it is known that it passes through the point A(-3;2) and is parallel to the straight line y=-4x.

The function equation has two unknown parameters: k and b. Therefore, the text of the problem must contain two conditions characterizing the graph of the function.

a) From the fact that the graph of the function is parallel to the straight line y=-4x, it follows that k=-4. That is, the function equation has the form

b) We just have to find b. It is known that the graph of the function passes through point A(-3;2). If a point belongs to the graph of a function, then when substituting its coordinates into the equation of the function, we obtain the correct equality:

hence b=-10

Thus, we need to plot the function

We know point A(-3;2), let’s take point B(0;-10)

Let's put these points in the coordinate plane and connect them with a straight line:

2. Write the equation of the line passing through the points A(1;1); B(2;4).

If a line passes through points with given coordinates, therefore, the coordinates of the points satisfy the equation of the line. That is, if we substitute the coordinates of the points into the equation of a straight line, we will get the correct equality.

Let's substitute the coordinates of each point into the equation and get a system of linear equations.

Subtract the first from the second equation of the system and get . Let's substitute the value of k into the first equation of the system and get b=-2.

So, the equation of the line.

3. Graph the Equation

To find at what values ​​of the unknown the product of several factors equals zero, you need to equate each factor to zero and take into account each multiplier.

This equation has no restrictions on ODZ. Let's factorize the second bracket and set each factor equal to zero. We obtain a set of equations:

Let's construct graphs of all equations of the set in one coordinate plane. This is the graph of the equation :

4. Construct a graph of the function if it is perpendicular to the line and passes through the point M(-1;2)

We will not build a graph, we will only find the equation of the line.

a) Since the graph of a function, if it is perpendicular to a line, therefore, hence. That is, the function equation has the form

b) We know that the graph of the function passes through the point M(-1;2). Let's substitute its coordinates into the equation of the function. We get:

From here.

Therefore, our function looks like: .

5. Graph the Function

Let's simplify the expression on the right side of the function equation.

Important! Before simplifying the expression, let's find its ODZ.

The denominator of a fraction cannot be zero, so title="x1">, title="x-1">.!}

Then our function takes the form:

Title="delim(lbrace)(matrix(3)(1)((y=x+2) (x1) (x-1)))( )">!}

That is, we need to build a graph of the function and cut out two points on it: with abscissas x=1 and x=-1:

Instructions

There are several ways to solve linear functions. Let's list the most of them. Most often used step by step method substitutions. In one of the equations it is necessary to express one variable in terms of another and substitute it into another equation. And so on until only one variable remains in one of the equations. To solve it, you need to leave a variable on one side of the equal sign (it can be with a coefficient), and on the other side of the equal sign all the numerical data, not forgetting to change the sign of the number to the opposite one when transferring. Having calculated one variable, substitute it into other expressions and continue calculations using the same algorithm.

For example, let's take a linear system functions, consisting of two equations:
2x+y-7=0;
x-y-2=0.
It is convenient to express x from the second equation:
x=y+2.
As you can see, when transferring from one part of the equality to another, the sign of y and variables changed, as described above.
We substitute the resulting expression into the first equation, thus excluding the variable x from it:
2*(y+2)+y-7=0.
Expanding the brackets:
2y+4+y-7=0.
We put together variables and numbers and add them up:
3у-3=0.
Move to right side equations, change the sign:
3y=3.
Divide by the total coefficient, we get:
y=1.
We substitute the resulting value into the first expression:
x=y+2.
We get x=3.

Another way to solve similar ones is to add two equations term by term to get a new one with one variable. The equation can be multiplied by a certain coefficient, the main thing is to multiply each member of the equation and not forget, and then add or subtract one equation from. This method is very economical when finding a linear functions.

Let’s take the already familiar system of equations with two variables:
2x+y-7=0;
x-y-2=0.
It is easy to notice that the coefficient of the variable y is identical in the first and second equations and differs only in sign. This means that when we add these two equations term by term, we get a new one, but with one variable.
2x+x+y-y-7-2=0;
3x-9=0.
We transfer numerical data to right side equations, changing the sign:
3x=9.
Finding the common factor equal to the coefficient, standing at x and divide both sides of the equation by it:
x=3.
The result can be substituted into any of the system equations to calculate y:
x-y-2=0;
3-у-2=0;
-y+1=0;
-y=-1;
y=1.

You can also calculate data by creating an accurate graph. To do this you need to find zeros functions. If one of the variables is equal to zero, then such a function is called homogeneous. Having solved such equations, you will get two points necessary and sufficient to construct a straight line - one of them will be located on the x-axis, the other on the y-axis.

We take any equation of the system and substitute the value x=0 there:
2*0+y-7=0;
We get y=7. Thus, the first point, let's call it A, will have coordinates A(0;7).
In order to calculate a point lying on the x-axis, it is convenient to substitute the value y=0 into the second equation of the system:
x-0-2=0;
x=2.
The second point (B) will have coordinates B (2;0).
We mark the obtained points on the coordinate grid and draw a straight line through them. If you plot it fairly accurately, other values ​​of x and y can be calculated directly from it.

1) Area function definitions and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Function zero is the value of the argument at which the value of the function is equal to zero.

3) Intervals of constant sign of a function.

Intervals of constant sign of a function are sets of argument values ​​on which the function values ​​are only positive or only negative.

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the ordinate.

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). Schedule odd function symmetrical about the origin.

6) Limited and unlimited functions.

A function is called bounded if there is such a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphs. Application of functions in economics.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function is called a function of the form , where x is a variable, a and b are real numbers.

Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.

Properties of a Linear Function

1. Domain of definition - the set of all real numbers: D(y)=R

2. The set of values ​​is the set of all real numbers: E(y)=R

3. The function takes a zero value when or.

4. The function increases (decreases) over the entire domain of definition.

5. Linear function continuous over the entire domain of definition, differentiable and .

2. Quadratic function.

A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic

Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember general rules, by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, derivative exponential equation, described. The calculations presented in next steps, will be based on the methods described therein.

Learn to distinguish between tasks in which slope needs to be calculated through the derivative of the function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.

Take the derivative of the function given to you. There is no need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function. Take the derivative according to the methods outlined in the article mentioned above:

• Derivative:

Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of a function is equal to the slope at a certain point. In other words, f"(x) is the slope of the function at any point (x,f(x)). In our example:

• Find the slope of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2).
• Derivative of a function:
  • f ′ (x) = 4 x + 6 (\displaystyle f"(x)=4x+6)
• Substitute the value of the “x” coordinate of this point:
  • f ′ (x) = 4 (4) + 6 (\displaystyle f"(x)=4(4)+6)
• Find the slope:
• Slope function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2) is equal to 22.

If possible, check your answer on a graph. Remember that the slope cannot be calculated at every point. Differential calculus is considering complex functions and complex graphs, where the slope cannot be calculated at every point, and in some cases the points do not lie on the graphs at all. If possible, use a graphing calculator to check that the slope of the function you are given is correct. Otherwise, draw a tangent to the graph at the point given to you and think about whether the slope value you found matches what you see on the graph.

• The tangent will have the same slope as the graph of the function at a certain point. To draw a tangent at a given point, move left/right on the X axis (in our example, 22 values ​​to the right), and then up one on the Y axis. Mark the point, and then connect it to the point given to you. In our example, connect the points with coordinates (4,2) and (26,3).

The concept of a numerical function. Methods for specifying a function. Properties of functions.

A numeric function is a function that acts from one numeric space (set) to another numeric space (set).

Three main ways to define a function: analytical, tabular and graphical.

1. Analytical.

The method of specifying a function using a formula is called analytical. This method is the main one in the mat. analysis, but in practice it is not convenient.

2. Tabular method of specifying a function.

A function can be specified using a table containing the argument values ​​and their corresponding function values.

3. Graphical method of specifying a function.

A function y=f(x) is said to be given graphically if its graph is constructed. This method of specifying a function makes it possible to determine the function values ​​only approximately, since constructing a graph and finding the function values ​​on it is associated with errors.

Properties of a function that must be taken into account when constructing its graph:

1) The domain of definition of the function.

Domain of the function, that is, those values ​​that the argument x of the function F =y (x) can take.

2) Intervals of increasing and decreasing functions.

The function is called increasing on the interval under consideration, if a larger value of the argument corresponds to a larger value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1 > x 2, then y(x 1) > y(x 2).

The function is called decreasing on the interval under consideration, if a larger value of the argument corresponds to a smaller value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1< х 2 , то у(х 1) < у(х 2).

3) Function zeros.

The points at which the function F = y (x) intersects the abscissa axis (they are obtained by solving the equation y(x) = 0) are called zeros of the function.

4) Even and odd functions.

The function is called even, if for all argument values ​​from the scope

y(-x) = y(x).

The graph of an even function is symmetrical about the ordinate.

The function is called odd, if for all values ​​of the argument from the domain of definition

y(-x) = -y(x).

The graph of an even function is symmetrical about the origin.

Many functions are neither even nor odd.

5) Periodicity of the function.

The function is called periodic, if there is a number P such that for all values ​​of the argument from the domain of definition

y(x + P) = y(x).

Linear function, its properties and graph.

A linear function is a function of the form y = kx + b, defined on the set of all real numbers.

k– slope (real number)

b– dummy term (real number)

x– independent variable.

· In the special case, if k = 0, we obtain a constant function y = b, the graph of which is a straight line parallel to the Ox axis passing through the point with coordinates (0; b).

· If b = 0, then we get the function y = kx, which is direct proportionality.

o Geometric meaning coefficient b is the length of the segment cut off by the straight line along the Oy axis, counting from the origin.

o The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, calculated counterclockwise.

Properties of a linear function:

1) The domain of definition of a linear function is the entire real axis;

2) If k ≠ 0, then the range of values ​​of the linear function is the entire real axis.

If k = 0, then the range of values ​​of the linear function consists of the number b;

3) Evenness and oddness of a linear function depend on the values ​​of the coefficients k and b.

a) b ≠ 0, k = 0, therefore, y = b – even;

b) b = 0, k ≠ 0, therefore y = kx – odd;

c) b ≠ 0, k ≠ 0, therefore y = kx + b is a function general view;

d) b = 0, k = 0, therefore y = 0 is both an even and an odd function.

4) A linear function does not have the property of periodicity;

5) Points of intersection with coordinate axes:

Ox: y = kx + b = 0, x = -b/k, therefore (-b/k; 0) is the point of intersection with the x-axis.

Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the ordinate.

Comment. If b = 0 and k = 0, then the function y = 0 vanishes for any value of the variable x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any value of the variable x.

6) The intervals of constancy of sign depend on the coefficient k.

a) k > 0; kx + b > 0, kx > -b, x > -b/k.

y = kx + b – positive at x from (-b/k; +∞),

y = kx + b – negative for x from (-∞; -b/k).

b)k< 0; kx + b < 0, kx < -b, x < -b/k.

y = kx + b – positive at x from (-∞; -b/k),

y = kx + b – negative for x of (-b/k; +∞).

c) k = 0, b > 0; y = kx + b is positive throughout the entire domain of definition,

k = 0, b< 0; y = kx + b отрицательна на всей области определения.

7) The monotonicity intervals of a linear function depend on the coefficient k.

k > 0, therefore y = kx + b increases throughout the entire domain of definition,

k< 0, следовательно y = kx + b убывает на всей области определения.

11. Function y = ax 2 + bx + c, its properties and graph.

The function y = ax 2 + bx + c (a, b, c are constants, a ≠ 0) is called quadratic In the simplest case, y = ax 2 (b = c = 0) the graph is a curved line passing through the origin. The curve serving as a graph of the function y = ax 2 is a parabola. Every parabola has an axis of symmetry called the axis of the parabola. The point O of the intersection of a parabola with its axis is called the vertex of the parabola.

The graph can be constructed according to the following scheme: 1) Find the coordinates of the vertex of the parabola x 0 = -b/2a; y 0 = y(x 0). 2) We construct several more points that belong to the parabola; when constructing, we can use the symmetries of the parabola relative to the straight line x = -b/2a. 3) Connect the indicated points with a smooth line. Example. Graph the function b = x 2 + 2x - 3. Solutions. The graph of the function is a parabola, the branches of which are directed upward. The abscissa of the vertex of the parabola x 0 = 2/(2 ∙1) = -1, its ordinates y(-1) = (1) 2 + 2(-1) - 3 = -4. So, the vertex of the parabola is point (-1; -4). Let's compile a table of values ​​for several points that are located to the right of the axis of symmetry of the parabola - straight line x = -1. Function properties.